Digital-electronics 简明教程
Digital Electronics - Threshold Logic
在前面的章节中，我们已经使用逻辑门实现了各种组合电路。除了非门之外，其余所有逻辑门至少有两个输入和一个输出。类似地 threshold gate 也至少包含一个输入和一个输出。
In previous chapters, we have implemented various combinational circuits using logic gates. Except NOT gate, the remaining all logic gates have at least two inputs and single output. Similarly, the threshold gate also contains at least one input and only one output.
此外，它包含每个输入的相应权重和一个阈值。这些权重和阈值的值可以是任何有限实数。
Additionally, it contains the respective weights to each input and a threshold value. The values of these weights and threshold could be of any finite real number.
Basics of Threshold gate
令阈值门的输入为 X1、X2、X3、…、Xn。这些输入的相应权重为 W1、W2、W3、…、Wn。阈值门的 symbol 如以下图形所示。
Let the inputs of threshold gate are X1, X2,X3,…, Xn. The corresponding weights of these inputs are W1, W2,W3, …, Wn. The symbol of Threshold gate is shown in the following figure.
Threshold gate 表示为一个圆，它有“n”个输入 X1 到 Xn 和单个输出 Y。该圆被分成两部分。一部分表示输入对应的权重，而另一部分表示阈值 T。
Threshold gate is represented with a circle and it is having 'n' inputs, X1 to Xn and single output, Y. This circle is made into two parts. One part represents the weights corresponding to the inputs and other part represents Threshold value, T.
具有相应权重的输入乘积的总和称为 weighted sum 。如果该加权和大于或等于阈值 T，那么只有输出 Y 才等于 1。否则，输出 Y 将等于 0。
The sum of products of inputs with corresponding weights is known as weighted sum. If this weighted sum is greater than or equal to Threshold value, T then only the output, Y will be equal to one. Otherwise, the output, Y will be equal to zero.
Mathematically ，我们可以在下面将输入和阈值门输出之间的这种关系写成如下形式。
Mathematically, we can write this relationship between inputs and output of Threshold gate as below.
如果 W1X1 + W2X2 + W3X3 + … + WnXn ≥ T，则 Y = 1。
\mathrm{Y \: = \: 1 \:\: if \: \: W_{1}X_{1} \: + \: W_{2}X_{2} \: + \: W_{3}X_{3} \: + \: \dotso \: + \: W_{n}X_{n} \: \geq \: T}
否则，Y = 0。
𝑌 = 0, otherwise.
因此，我们可以仅通过更改权重和/或阈值 T 的值来实现各种逻辑门和布尔函数。
Therefore, we can implement various logic gates and Boolean functions just by changing the values of weights and / or Threshold value, T.
Example
让我们为以下阈值门找到 simplified Boolean function 。
Let us find the simplified Boolean function for the following Threshold gate.
该阈值门具有三个输入 X1、X2、X3 和一个输出 Y。
This Threshold gate is having three inputs X1, X2, X3 and one output Y.
分别对应于输入 X1、X2 和 X3 的权重为 W1 = 2、W2 = 1 和 W3 = -4。
The weights corresponding to the inputs X1, X2 & X3 are W1 = 2, W2 = 1 & W3 = -4 respectively.
阈值门的值为 T = -1。
The value of Threshold gate is T = -1.
阈值门的 weighted sum 是
The weighted sum of Threshold gate is
W = W1X1 + W2X2 + W3X3
\mathrm{W \: = \: W_{1}X_{1} \: + \: W_{2}X_{2} \: + \: W_{3}X_{3}}
在上述方程中代入给定的权重。
Substitute the given weights in the above equation.
⇒ W = 2X1 + X2 − 4X3
\mathrm{\Rightarrow \: W \: = \: 2X_{1} \: + \: X_{2} \: − \: 4X_{3}}
如果 W ≥ −1，则阈值门的输出 Y 将为“1”，否则将为“0”。
Output of Threshold gate, Y will be '1' if W ≥ −1, otherwise it will be '0'.
以下 table 显示了所有可能输入组合的输入和输出之间的关系。
The following table shows the relationship between the input and output for all possible combination of inputs.
Input |
Weighted Sum |
Output |
X1 |
X2 |
X3 |
W = 2X1 + X2 - 4X3 |
Y |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
-4 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
-3 |
0 |
1 |
0 |
0 |
2 |
1 |
1 |
0 |
1 |
-2 |
0 |
1 |
1 |
0 |
3 |
1 |
1 |
1 |
1 |
-1 |
1 |
从上表中，我们可以为输出 Y 编写 Boolean function ，如下所示：
From the above table, we can write the Boolean function for output, Y as
Y = Σm(0、2、4、6、7)
\mathrm{Y \: = \: \sum m( 0,2,4,6,7)}
使用 3 variable K-Map 简化此布尔函数的过程如下图所示。
The simplification of this Boolean function using 3 variable K-Map is shown in the following figure.
因此，给定阈值门的 simplified Boolean function 为 Y = X3' + X1 X2。
Therefore, the simplified Boolean function for given Threshold gate is Y = X3' + X1 X2.
Synthesis of Threshold Functions
阈值门也称为 universal gate ，因为我们可以使用阈值门来实现任何布尔函数。有时候，可能无法使用单个阈值门来实现一些逻辑门和布尔函数。在那种情况下，我们可能需要多个阈值门。
Threshold gate is also called as universal gate because we can implement any Boolean function using Threshold gate(s). Some-times, it may not possible to implement few logic gates and Boolean functions by using single Threshold gate. In that case, we may require multiple Threshold gates.
按照 steps 的这些步骤使用单个阈值门实现布尔函数。
Follow these steps for implementing a Boolean function using single Threshold gate.
Step 1 - 为给定的布尔函数制定一个 Truth table 表格。
Step 1 − Formulate a Truth table for given Boolean function.
Step 2 - 在上面的真值表中，添加（包含）一列，给出 weighted sums 和 Threshold value 之间的关系。
Step 2 − In the above Truth table, add (include) one more column, which gives the relation between weighted sums and Threshold value.
Step 3 - 对于每种输入组合，按照下面提到的方法，写出自重和与阈值之间的关系。
Step 3 − Write the relation between weighted sums and threshold for each combination of inputs as mentioned below.
-
If the output of Boolean function is 1, then the weighted sum will be greater than or equal to Threshold value for those combination of inputs.
-
If the output of Boolean function is 0, then the weighted sum will be less than Threshold value for those combination of inputs.
Step 4 - 以这样一种方式选择权重和阈值，使它们满足上述表格最后一列中出现的所有关系。
Step 4 − Choose the values of weights & Threshold in such a way that they should satisfy all the relations present in last column of the above table.
Step 5 - 绘制带有这些权重和阈值的阈值门的 symbol 。
Step 5 − Draw the symbol of Threshold gate with those weights and Threshold value.
Example
让我们使用单个阈值门来实现下面的 Boolean function 。
Let us implement the following Boolean function using single Threshold gate.
\(\mathrm{Y( X_{1},X_{2},X_{3})\:=\: \sum m ( 0,2,4,6,7)}\)
\mathrm{Y( X_{1},X_{2},X_{3})\:=\: \sum m ( 0,2,4,6,7)}
给定的布尔函数是三变量函数，以最小项和的形式表示。此函数的 Truth table 如下所示。
The given Boolean function is a three variable function, which is represented in sum of min terms form. The Truth table of this function is shown below.
Input |
Output |
X1 |
X2 |
X3 |
Y |
0 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
0 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
现在，让我们在上面的真值表中添加（包含）一列。此最后一列包含对于每种输入组合的 weighted sums (W) and Threshold 值 (T) 之间的关系。
Now, let us add (include) one more column to the above Truth table. This last column contains the relations between weighted sums (W) and Threshold value (T) for each combination of inputs.
Input |
Output |
Relations Between W & T |
X1 |
X2 |
X3 |
Y |
0 |
0 |
0 |
1 |
$\mathrm{0 \: \geq \: T}$ |
0 |
0 |
1 |
0 |
$\mathrm{W_{3} \: \lt \: T}$ |
0 |
1 |
0 |
1 |
$\mathrm{W_{2} \: \geq \: T}$ |
0 |
1 |
1 |
0 |
$\mathrm{W_{2}+W_{3} \: \lt \: T}$ |
1 |
0 |
0 |
1 |
$\mathrm{W_{1} \: \geq \: T}$ |
1 |
0 |
1 |
0 |
$\mathrm{W_{1}+W_{3} \: \lt \: T}$ |
1 |
1 |
0 |
1 |
$\mathrm{W_{1}+W_{2} \: \geq \: T}$ |
1 |
1 |
1 |
以下为上述表格的结论。
Following are the conclusions from the above table.
-
The value of Threshold should be either zero or negative based on first relation.
-
The value of W3 should be negative based on first and second relations.
-
The values of W1 and W2 should be greater than or equal Threshold value based on fifth and third relations.
-
W2 should be greater than W3 based on fourth relation.
我们可以根据上述结论为权重和阈值选择以下值。
We can choose the following values for weights and Threshold based on the above conclusions.
W1 = 2, W2 = 1, W3 = -4 和 T = -1
W1 = 2, W2 = 1, W3 = -4 & T = -1
具有上述值的阈值门的 symbol 如下所示。
The symbol of Threshold gate with the above values is shown below.
因此，该阈值门实现了给定的 Boolean function , Y(X1,X2,X3):=∑m(0,2,4,6,7)。
Therefore, this Threshold gate implements the given Boolean function, $\mathrm{Y( X_{1}, X_{2},X_{3}) \:=\: \sum m (0,2,4,6,7)}$.